In a sentence where "write" doesn't have a direct object, "write you" is American English. In British English it's "write to you":
- I'd like to write to you about it (BrE) (and sometimes AmE, I think).
- I'd like to write you about it (AmE only).
However, when "write" has a direct object (for example, 'a poem'), the most usual form, in any variant of English, is:
- I'd like to write you a poem about it. (Here, "write you a poem" means "write a poem for you".)
I'm not sure what you mean by "a bit", sunny, (briefly?, a short piece of text?, ...) — but "I'd like to write (to) you a bit about it" isn't something I'd expect a native speaker to say.